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I am reading "Introduction to the Foundation of Mathematics" ,by R L.Wilder(2nd Ed.), where Mortiz Pasch's works are described in paragraph 1.5. There is a quotation in that paragraph-

" For if,on replacing the geometric terms in the basic set of propositions by certain other terms,true propositions are obtained,then corresponding replacements may be made in the theorem;"

My questions are as follows
1) Can certain other terms be arithmetical terms or algebraic terms also? 2) Either the terms,used for replacement,or the conclusion after replacement - Which one will be transformed into the theorem?

Shubhadip Ghosh
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1 Answers1

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It is the fundamental definition of model of a theory.

We have that the theorems of geometry are logical consequences of the axioms of geometry, i.e. the theorems are satisfied in every interpretation which satisfy the axioms.

Thus, if we replace the "geometric terms", like : point, line (and relations, like : between) used into the "basic propositions" (the axioms) with oher terms (e.g. couple of real numbers, and so on) and we find that the "translated" axioms are proposition which are true in the new interpretation (the new "domain of discourse), we can replace consistenly the geometrical terms in a geometrical theorem whatever with the new terms, and the result will be a true proposition of the new interpretation.

Mauro ALLEGRANZA
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